{"paper":{"title":"On asymptotic dynamics for $L^2$ critical generalized KdV equations with a saturated perturbation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Yang Lan","submitted_at":"2016-09-16T17:21:57Z","abstract_excerpt":"In this paper, we consider the $L^2$ critical gKdV equation with a saturated perturbation: $\\partial_t u+(u_{xx}+u^5-\\gamma u|u|^{q-1})_x=0$, where $q>5$ and $0<\\gamma\\ll1$. For any initial data $u_0\\in H^1$, the corresponding solution is always global and bounded in $H^1$. This equation has a family of solitons, and our goal is to classify the dynamics near soliton. Together with a suitable decay assumption, there are only three possibilities: (i) the solution converges asymptotically to a solitary wave, whose $H^1$ norm is of size $\\gamma^{-2/(q-1)}$, as $\\gamma\\rightarrow0$; (ii) the soluti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.05146","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}